Determine whether each function graphed or defined is one-to-one. y = 5|x+2|

Verified step by step guidance

1. Understand the definition of a one-to-one function: A function is one-to-one if each output value is paired with exactly one input value, meaning it passes the horizontal line test.
2. Consider the given function: \( y = 5|x+2| \). This is an absolute value function, which typically has a 'V' shape when graphed.
3. Graph the function \( y = 5|x+2| \). The vertex of the graph is at \( x = -2 \), and the graph opens upwards with a slope of 5 on either side of the vertex.
4. Apply the horizontal line test: Draw horizontal lines across the graph. If any horizontal line intersects the graph at more than one point, the function is not one-to-one.
5. Observe that for \( y = 5|x+2| \), horizontal lines will intersect the graph at two points for any \( y > 0 \), indicating that the function is not one-to-one.

Key Concepts

One-to-One Function

A one-to-one function is a type of function where each output value is associated with exactly one input value. This means that no two different inputs produce the same output. To determine if a function is one-to-one, one can use the horizontal line test: if any horizontal line intersects the graph of the function more than once, the function is not one-to-one.

Absolute Value Function

An absolute value function is defined as f(x) = |x|, which outputs the non-negative value of x. The graph of an absolute value function is V-shaped, with a vertex at the origin. In the case of the function y = 5|x + 2|, the graph is vertically stretched by a factor of 5 and shifted left by 2 units, affecting its symmetry and one-to-one nature.

Graphical Analysis

Graphical analysis involves examining the visual representation of a function to understand its properties, such as continuity, intercepts, and whether it is one-to-one. By analyzing the shape and behavior of the graph, one can apply tests like the horizontal line test to determine if the function meets specific criteria, such as being one-to-one.